limit as x approaches-2 of x/((x+2)^2)

{ "query": { "display": "$$\\lim_{x\\to\\:-2}\\left(\\frac{x}{\\left(x+2\\right)^{2}}\\right)$$", "symbolab_question": "BIG_OPERATOR#\\lim _{x\\to -2}(\\frac{x}{(x+2)^{2}})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Limits", "subTopic": "SingleVar", "default": "-\\infty ", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\lim_{x\\to\\:-2}\\left(\\frac{x}{\\left(x+2\\right)^{2}}\\right)=-\\infty\\:$$", "input": "\\lim_{x\\to\\:-2}\\left(\\frac{x}{\\left(x+2\\right)^{2}}\\right)", "steps": [ { "type": "step", "primary": "If $$\\lim_{x\\to{a-}}{f\\left(x\\right)}=\\lim_{x\\to{a+}}{f\\left(x\\right)}=L$$ then $$\\lim_{x\\to{a}}{f\\left(x\\right)}=L$$" }, { "type": "interim", "title": "$$\\lim_{x\\to\\:-2-}\\left(\\frac{x}{\\left(x+2\\right)^{2}}\\right)=-\\infty\\:$$", "input": "\\lim_{x\\to\\:-2-}\\left(\\frac{x}{\\left(x+2\\right)^{2}}\\right)", "steps": [ { "type": "step", "primary": "Apply the following algebraic property$$:{\\quad}a+b=a\\left(1+\\frac{b}{a}\\right)$$<br/>$$\\frac{x}{\\left(x+2\\right)^{2}}=\\frac{x}{\\left(x\\left(1+\\frac{2}{x}\\right)\\right)^{2}}$$", "result": "=\\lim_{x\\to\\:-2-}\\left(\\frac{x}{\\left(x\\left(1+\\frac{2}{x}\\right)\\right)^{2}}\\right)" }, { "type": "interim", "title": "Simplify $$\\frac{x}{\\left(x\\left(1+\\frac{2}{x}\\right)\\right)^{2}}:{\\quad}\\frac{1}{\\frac{4}{x}+4+x}$$", "input": "\\frac{x}{\\left(x\\left(1+\\frac{2}{x}\\right)\\right)^{2}}", "steps": [ { "type": "interim", "title": "Factor $$\\left(x\\left(\\frac{2}{x}+1\\right)\\right)^{2}:{\\quad}x^{2}\\left(\\frac{2}{x}+1\\right)^{2}$$", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(ab\\right)^{c}=a^{c}b^{c}$$", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "result": "=x^{2}\\left(\\frac{2}{x}+1\\right)^{2}" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{x}{x^{2}\\left(\\frac{2}{x}+1\\right)^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{1}{x\\left(\\frac{2}{x}+1\\right)^{2}}" }, { "type": "interim", "title": "Expand $$x\\left(\\frac{2}{x}+1\\right)^{2}:{\\quad}\\frac{4}{x}+4+x$$", "input": "x\\left(\\frac{2}{x}+1\\right)^{2}", "result": "=\\frac{1}{\\frac{4}{x}+4+x}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{2}{x}+1\\right)^{2}:{\\quad}\\frac{4}{x^{2}}+\\frac{4}{x}+1$$", "result": "=x\\left(\\frac{4}{x^{2}}+\\frac{4}{x}+1\\right)", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a+b\\right)^{2}=a^{2}+2ab+b^{2}$$", "secondary": [ "$$a=\\frac{2}{x},\\:\\:b=1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=\\left(\\frac{2}{x}\\right)^{2}+2\\cdot\\:\\frac{2}{x}\\cdot\\:1+1^{2}" }, { "type": "interim", "title": "Simplify $$\\left(\\frac{2}{x}\\right)^{2}+2\\cdot\\:\\frac{2}{x}\\cdot\\:1+1^{2}:{\\quad}\\frac{4}{x^{2}}+\\frac{4}{x}+1$$", "input": "\\left(\\frac{2}{x}\\right)^{2}+2\\cdot\\:\\frac{2}{x}\\cdot\\:1+1^{2}", "result": "=\\frac{4}{x^{2}}+\\frac{4}{x}+1", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\left(\\frac{2}{x}\\right)^{2}+2\\cdot\\:1\\cdot\\:\\frac{2}{x}+1" }, { "type": "interim", "title": "$$\\left(\\frac{2}{x}\\right)^{2}=\\frac{4}{x^{2}}$$", "input": "\\left(\\frac{2}{x}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{2^{2}}{x^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\frac{4}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZJbQQB/d5WyHfvqLuvSsHI5IpdliG1E4K4EtDGLN9yvMwViaLUXkeD+JukROhWdjHz1v/wzVPirQNs1yHWg28/8//6/nV5O4fb8Xgwi7mapwNWwK6RZNsGLG7ciUYsGJ3RQTWnXpwJXSdpvAWh4snViVI3uvN1by+AN9NfjoKFU=" } }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{2}{x}\\cdot\\:1=\\frac{4}{x}$$", "input": "2\\cdot\\:\\frac{2}{x}\\cdot\\:1", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=1\\cdot\\:\\frac{2\\cdot\\:2}{x}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{4}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qviuGzXpksjtbtgDPqsTbv9LpV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAlvTW8J6Ksi2sNNwou0Q/ozu2RJ3rpj7pu1O/b/KBze535IdS4PVmAL+EtN/2VV0x4TGs8945Vq7f7zQJklxf7aCwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{4}{x^{2}}+\\frac{4}{x}+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Expand $$x\\left(\\frac{4}{x^{2}}+\\frac{4}{x}+1\\right):{\\quad}\\frac{4}{x}+4+x$$", "input": "x\\left(\\frac{4}{x^{2}}+\\frac{4}{x}+1\\right)", "result": "=\\frac{4}{x}+4+x", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=x\\frac{4}{x^{2}}+x\\frac{4}{x}+x\\cdot\\:1", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "result": "=\\frac{4}{x^{2}}x+\\frac{4}{x}x+1\\cdot\\:x" }, { "type": "interim", "title": "Simplify $$\\frac{4}{x^{2}}x+\\frac{4}{x}x+1\\cdot\\:x:{\\quad}\\frac{4}{x}+4+x$$", "input": "\\frac{4}{x^{2}}x+\\frac{4}{x}x+1\\cdot\\:x", "result": "=\\frac{4}{x}+4+x", "steps": [ { "type": "interim", "title": "$$\\frac{4}{x^{2}}x=\\frac{4}{x}$$", "input": "\\frac{4}{x^{2}}x", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{4x}{x^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{4}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wC5as4rb80hIhEB6CK89Yt4k0OFdoisEPoiIM5z/e+WjkVi15I8rBefLi4Iyt2wr+HFNia1f2SXxhAEhgx+8TQeTV4u20zW11V5PLOwL6WtwASL/CcrB6jQPzgIobvKrHsfg1lpsjRsQH/JwOuGayw==" } }, { "type": "interim", "title": "$$\\frac{4}{x}x=4$$", "input": "\\frac{4}{x}x", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{4x}{x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p0Gjue7IO4Kd0HX30crCsC061ljBSPJeENOw2efoSWuqze2Yzn0Gom7AYUa+PIjkR4IEq5gqBo0nbneAsjr1TlISDkKydAMaY+4YzU0gOW0=" } }, { "type": "interim", "title": "$$1\\cdot\\:x=x$$", "input": "1\\cdot\\:x", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ASx2YODBupsHY/9yrO15bd13jtrSFDx+UNsawjlOjV3pfPCe8nQAZY1bE89UDVgMPJrYhwc+zvuHrOLz58Ml2oD661lPR3w/W4zyCV9dwUw=" } }, { "type": "step", "result": "=\\frac{4}{x}+4+x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HSNp0b0fUkUA7vbkWvx1cL6JWSz+y451XMbjw8jACaVVGM30su7exDI16VkBN/EAzMFYmi1F5Hg/ibpEToVnY6dOUeeuFjGLMO4rdTNSBtw2eVx9hLE8iCynUUflNNBg/G2ANG+ldkUkzmezrPTTg9ZWuscTyUNTipqsWI9f2Qi+iVks/suOdVzG48PIwAml2T7z9li4FwHFKCqHzAPKZg==" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nmViRmUF4HwQakq3zOY0Xp9wuFWY2iSOqJSsGXMs9rcJQJZuTAY5js+oqjdT8kslxyebD2dMHPt88v9z+vwIQ/pY5r/NYN14WD2SF6U6tTwezFilETfuNygjs0XPkV2hgNG+61vKRzPJFG3mJB8N1lkXwmHxkVMNXXBBq/WB+wM=" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\lim_{x\\to\\:-2-}\\left(\\frac{1}{\\frac{4}{x}+4+x}\\right)" }, { "type": "step", "primary": "For $$x\\:$$approaching $$-2\\:$$from the left$$,\\:x<-2\\quad\\Rightarrow\\quad\\:\\frac{4}{x}+4+x<0$$", "secondary": [ "The denominator is a negative quantity approaching 0 from the left" ], "result": "=-\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:-2+}\\left(\\frac{x}{\\left(x+2\\right)^{2}}\\right)=-\\infty\\:$$", "input": "\\lim_{x\\to\\:-2+}\\left(\\frac{x}{\\left(x+2\\right)^{2}}\\right)", "steps": [ { "type": "step", "primary": "Apply the following algebraic property$$:{\\quad}a+b=a\\left(1+\\frac{b}{a}\\right)$$<br/>$$\\frac{x}{\\left(x+2\\right)^{2}}=\\frac{x}{\\left(x\\left(1+\\frac{2}{x}\\right)\\right)^{2}}$$", "result": "=\\lim_{x\\to\\:-2+}\\left(\\frac{x}{\\left(x\\left(1+\\frac{2}{x}\\right)\\right)^{2}}\\right)" }, { "type": "interim", "title": "Simplify $$\\frac{x}{\\left(x\\left(1+\\frac{2}{x}\\right)\\right)^{2}}:{\\quad}\\frac{1}{\\frac{4}{x}+4+x}$$", "input": "\\frac{x}{\\left(x\\left(1+\\frac{2}{x}\\right)\\right)^{2}}", "steps": [ { "type": "interim", "title": "Factor $$\\left(x\\left(\\frac{2}{x}+1\\right)\\right)^{2}:{\\quad}x^{2}\\left(\\frac{2}{x}+1\\right)^{2}$$", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(ab\\right)^{c}=a^{c}b^{c}$$", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "result": "=x^{2}\\left(\\frac{2}{x}+1\\right)^{2}" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{x}{x^{2}\\left(\\frac{2}{x}+1\\right)^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{1}{x\\left(\\frac{2}{x}+1\\right)^{2}}" }, { "type": "interim", "title": "Expand $$x\\left(\\frac{2}{x}+1\\right)^{2}:{\\quad}\\frac{4}{x}+4+x$$", "input": "x\\left(\\frac{2}{x}+1\\right)^{2}", "result": "=\\frac{1}{\\frac{4}{x}+4+x}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{2}{x}+1\\right)^{2}:{\\quad}\\frac{4}{x^{2}}+\\frac{4}{x}+1$$", "result": "=x\\left(\\frac{4}{x^{2}}+\\frac{4}{x}+1\\right)", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a+b\\right)^{2}=a^{2}+2ab+b^{2}$$", "secondary": [ "$$a=\\frac{2}{x},\\:\\:b=1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=\\left(\\frac{2}{x}\\right)^{2}+2\\cdot\\:\\frac{2}{x}\\cdot\\:1+1^{2}" }, { "type": "interim", "title": "Simplify $$\\left(\\frac{2}{x}\\right)^{2}+2\\cdot\\:\\frac{2}{x}\\cdot\\:1+1^{2}:{\\quad}\\frac{4}{x^{2}}+\\frac{4}{x}+1$$", "input": "\\left(\\frac{2}{x}\\right)^{2}+2\\cdot\\:\\frac{2}{x}\\cdot\\:1+1^{2}", "result": "=\\frac{4}{x^{2}}+\\frac{4}{x}+1", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\left(\\frac{2}{x}\\right)^{2}+2\\cdot\\:1\\cdot\\:\\frac{2}{x}+1" }, { "type": "interim", "title": "$$\\left(\\frac{2}{x}\\right)^{2}=\\frac{4}{x^{2}}$$", "input": "\\left(\\frac{2}{x}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{2^{2}}{x^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\frac{4}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZJbQQB/d5WyHfvqLuvSsHI5IpdliG1E4K4EtDGLN9yvMwViaLUXkeD+JukROhWdjHz1v/wzVPirQNs1yHWg28/8//6/nV5O4fb8Xgwi7mapwNWwK6RZNsGLG7ciUYsGJ3RQTWnXpwJXSdpvAWh4snViVI3uvN1by+AN9NfjoKFU=" } }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{2}{x}\\cdot\\:1=\\frac{4}{x}$$", "input": "2\\cdot\\:\\frac{2}{x}\\cdot\\:1", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=1\\cdot\\:\\frac{2\\cdot\\:2}{x}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{4}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qviuGzXpksjtbtgDPqsTbv9LpV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAlvTW8J6Ksi2sNNwou0Q/ozu2RJ3rpj7pu1O/b/KBze535IdS4PVmAL+EtN/2VV0x4TGs8945Vq7f7zQJklxf7aCwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{4}{x^{2}}+\\frac{4}{x}+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Expand $$x\\left(\\frac{4}{x^{2}}+\\frac{4}{x}+1\\right):{\\quad}\\frac{4}{x}+4+x$$", "input": "x\\left(\\frac{4}{x^{2}}+\\frac{4}{x}+1\\right)", "result": "=\\frac{4}{x}+4+x", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=x\\frac{4}{x^{2}}+x\\frac{4}{x}+x\\cdot\\:1", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "result": "=\\frac{4}{x^{2}}x+\\frac{4}{x}x+1\\cdot\\:x" }, { "type": "interim", "title": "Simplify $$\\frac{4}{x^{2}}x+\\frac{4}{x}x+1\\cdot\\:x:{\\quad}\\frac{4}{x}+4+x$$", "input": "\\frac{4}{x^{2}}x+\\frac{4}{x}x+1\\cdot\\:x", "result": "=\\frac{4}{x}+4+x", "steps": [ { "type": "interim", "title": "$$\\frac{4}{x^{2}}x=\\frac{4}{x}$$", "input": "\\frac{4}{x^{2}}x", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{4x}{x^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{4}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wC5as4rb80hIhEB6CK89Yt4k0OFdoisEPoiIM5z/e+WjkVi15I8rBefLi4Iyt2wr+HFNia1f2SXxhAEhgx+8TQeTV4u20zW11V5PLOwL6WtwASL/CcrB6jQPzgIobvKrHsfg1lpsjRsQH/JwOuGayw==" } }, { "type": "interim", "title": "$$\\frac{4}{x}x=4$$", "input": "\\frac{4}{x}x", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{4x}{x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7p0Gjue7IO4Kd0HX30crCsC061ljBSPJeENOw2efoSWuqze2Yzn0Gom7AYUa+PIjkR4IEq5gqBo0nbneAsjr1TlISDkKydAMaY+4YzU0gOW0=" } }, { "type": "interim", "title": "$$1\\cdot\\:x=x$$", "input": "1\\cdot\\:x", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ASx2YODBupsHY/9yrO15bd13jtrSFDx+UNsawjlOjV3pfPCe8nQAZY1bE89UDVgMPJrYhwc+zvuHrOLz58Ml2oD661lPR3w/W4zyCV9dwUw=" } }, { "type": "step", "result": "=\\frac{4}{x}+4+x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HSNp0b0fUkUA7vbkWvx1cL6JWSz+y451XMbjw8jACaVVGM30su7exDI16VkBN/EAzMFYmi1F5Hg/ibpEToVnY6dOUeeuFjGLMO4rdTNSBtw2eVx9hLE8iCynUUflNNBg/G2ANG+ldkUkzmezrPTTg9ZWuscTyUNTipqsWI9f2Qi+iVks/suOdVzG48PIwAml2T7z9li4FwHFKCqHzAPKZg==" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nmViRmUF4HwQakq3zOY0Xp9wuFWY2iSOqJSsGXMs9rcJQJZuTAY5js+oqjdT8kslxyebD2dMHPt88v9z+vwIQ/pY5r/NYN14WD2SF6U6tTwezFilETfuNygjs0XPkV2hgNG+61vKRzPJFG3mJB8N1lkXwmHxkVMNXXBBq/WB+wM=" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\lim_{x\\to\\:-2+}\\left(\\frac{1}{\\frac{4}{x}+4+x}\\right)" }, { "type": "step", "primary": "For $$x\\:$$approaching $$-2\\:$$from the right$$,\\:x>-2\\quad\\Rightarrow\\quad\\:\\frac{4}{x}+4+x<0$$", "secondary": [ "The denominator is a negative quantity approaching 0 from the left" ], "result": "=-\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=-\\infty\\:" } ], "meta": { "practiceLink": "/practice/limits-practice", "practiceTopic": "Limits" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }